Calculating the Fock matrix

Fock matrix elements are not actually calculated. The Fock matrix elements are not any definite physical quantities, but rather energy levels relative to a in units of ft, making them 0 or —1. One can try to estimate a and ft, but the SHM does not define them quantitatively. 

The program calculates the Fock matrix elements /(Eqs. 4.91 and 4.92) using stored values of ionization energies 7, the overlap integrals S, and the proportionality constant K of that particular program. The matrix elements are assembled into the Fock matrix H. 

Each element Grs is calculated from a density matrix element Ptu (Eqs. 5.80 and 5.81) and two two-electron integrals (rsjtu) and (ru/ts) (Eqs. 5.73 and 5.77). The required one-electron integrals for calculating the Fock matrix F are... 

Step 4 - Calculating the Fock matrix (a) The one-electron matrices From Eq. 5.100... 

Calculate the Fock matrix and then the transformed Fock matrix F. ... 

From an initial set of orthonormal MOs we calculate the Fock matrix according to (10.3.24). Diagonalization of the Fock matrix in this basis yields... 

In order to form the Fock matrix ofan ah iniiio calculation, all the... 

By replacing the superscripts a and (i by Pand tx, respectively, in th e above th ree eq u ation s. you can easily get three similar equations for the Fock matrix elements for beta orbitals. Similar expressions to the above for Fock matrix elements ol restricted Ilartree-Fock (RIIF) calculations can be generated by simply icplaeing 1- (or I P) by 1/2 P in the above equation s. 

In ub initio calculations all elements of the Fock matrix are calculated using Equation (2.226), ii re peifive of whether the basis functions ip, cp, formally bonded. To discuss the semi-empirical melh ids it is useful to consider the Fock matrix elements in three groups (the diagonal... 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions... 

Choose the DIIS SCF convergence accelerator to potentially speed up SCF convergence. DIIS often reduces the number of iterations required to reach a convergence limit. However, it takes memory to store the Fock matrices from the previous iterations and this option may increase the computational time for individual iterations because the Fock matrix has to be calculated as a linear combination of the current Fock matrix and Fock matrices from previous iterations. 

Thus, HyperChem occasionally uses a three-point interpolation of the density matrix to accelerate the convergence of quantum mechanics calculations when the number of iterations is exactly divisible by three and certain criteria are met by the density matrices. The interpolated density matrix is then used to form the Fock matrix used by the next iteration. This method usually accelerates convergent calculations. However, interpolation with the MINDO/3, MNDO, AMI, and PM3 methods can fail on systems that have a significant charge buildup. 

In order to form the Fock matrix of an ab initio calculation, all the core-Hamiltonian matrix elements, H y, and two-electron integrals (pvIXa) have to be computed. If the total number of basis functions is m, the total number of the core Hamiltonian matrix elements is... 

To construct the Fock matrix, eq. (3.51), integrals over all pairs of basis functions and the one-electron operator h are needed. For M basis functions there are of the order of of such one-electron integrals. These one-integrals are also known as core integrals, they describe the interaction of an electron with the whole frame of bare nuclei. The second part of the Fock matrix involves integrals over four basis functions and the g two-electron operator. There are of the order of of these two-electron integrals. In conventional HF methods the two-electron integrals are calculated and saved before the... 

After the HFW integrals have been assembled, we then move on to the self-consistent field (SCF) procedure. For the most part this is the same as the HF version (10), with the exception of constructing the Fock matrix. The Fock matrix elements for an unrestricted HFW calculation are analogous to their HF counterparts and are given by... 

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